# Tagged Questions

Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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### $\pi_0(SO(N))$ and $\pi_0(O(N))$: Inconsistency between Bott periodicity and basic understanding of $\pi_0$

I need to know the homotopy groups of the oriented Grassmannian $\widetilde{Gr}(\infty,\infty) \cong \lim_{N \rightarrow \infty} SO(2N)/(SO(N) \times SO(N))$, and I've run into an inconsistency. It ...
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### Counter-example for $\tilde{H} (X/A) \cong H (X, A)$?

Yo! I was not able to find a counter-example to $$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations $A \hookrightarrow X$ (or more generally whenever $A$ is a deformation ...
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### If two maps are homotopic, are the images homotopy equivalent?

My question is; If two continuous maps $f,g:X\rightarrow Y$ are homotopic, are the images $Im(f),Im(g)$ homotopy equivalent? Clearly, the converse is false. If it is false, is there any condition ...
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### Homotopy type of some lattices with top and bottom removed

There was an interesting question on MO which OP removed by some reason. Here is a (more or less) equivalent form. Take a finite cartesian product of finite linear orders, and remove top and bottom. ...
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### Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
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### Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...
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### basepoint problem: Is there an action of $\pi_1(B)$ on $\pi_1(F)$ for $F$ path connected

I am doing this to try to figure out The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$ . Let the fibration $F \hookrightarrow E \xrightarrow{p} B$ be ...
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### Is the nth homotopy group isomorphic to $[T^n, X]$

Following Spanier's book on algebraic topology chapter $1$, section $6$ about suspensions, I'm wondering about the following questions: 1) We know that $S^n$ is an $H$-cogroup for all $n\geq1$ ...
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### Let $G$ a simple connected topological group and $H$ a normal discrete subgroup, then $\pi_1(G/H,e) = H.$

I know that $G$ is a covering space for $G/H$ and there is a injection between the fundamental group of $G$ and $G/H.$ How to proceed to show that $\pi_1(G/H,e) = H?$.
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### bottom map of pullback square is cofibration $\Rightarrow$top map is cofibration

$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I am trying to show that for a given fibration $E \xrightarrow{p} B$, and a cell structure $\{B_n\}$ on $B$ that , that $p^{-1}B_n \to p^{-1}B_{n+1}$ is ...
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### Second homotopy of $S^1\vee S^2 \vee T^2$

How can I prove that the second homotopy group of $S^1\vee S^2 \vee T^2$ is infinitely generated? I know that the second homotopu groups of $S^2$ and $T^2$ are finitely generated, so is kind of ...
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### Homotopy of closed curves is also a closed curve?

I'm trying to prove the following statement: Let $\gamma_1$ and $\gamma_2$ be two closed curves from $[a, b]$ to $\mathbb{C}$, and let $h: [a,b]\times[0,1] \to \mathbb{C}$ be a homotopy between ...
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### multiplication in H-space and loopspace of the H-space

Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy. Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $\circ$ on ...
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### A question on the non trivial rank three bundle on $S^2$

We know that number of rank $3$ vector bundles on $S^2$ is just the number of equivalence classes of maps from $S^1$ to $SO(3)$. This implies that there is one and only one non trivial vector bundle ...
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### Is the sphere with a diameter homotopy equivalent to a surface?

This is for a homework problem: Take the unit sphere $\mathbb{S}^2$ and join the north and south poles with a line segment. Is the resulting space homotopy equivalent to a surface? Intuitively, ...
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### How to construct a homotopy equivalence between a mobius band and a circle?

A mobius band is homotopic equivalent to a circle because the mobius band can deformation retract onto a circle. I am wondering how could we understand this fact from the definition of being ...
When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
I'm looking for a clarification of the highlighted comment taken from "A User's Guide to Algebraic Topology" by Dodson & Parker. In order to make the setting clear, I uploaded definition $8.1.6$ ...